Is massively collaborative mathematics possible?

Of course, one might say, there are certain kinds of problems that lend themselves to huge collaborations. One has only to think of the proof of the classification of finite simple groups, or of a rather different kind of example such as a search for a new largest prime carried out during the downtime of thousands of PCs around the world. But my question is a different one. What about the solving of a problem that does not naturally split up into a vast number of subtasks? Are such problems best tackled by people for some that belongs to the set ? (Examples of famous papers with four authors do not count as an interesting answer to this question.)

It seems to me that, at least in theory, a different model could work: different, that is, from the usual model of people working in isolation or collaborating with one or two others. Suppose one had a forum (in the non-technical sense, but quite possibly in the technical sense as well) for the online discussion of a particular problem. The idea would be that anybody who had anything whatsoever to say about the problem could chip in. And the ethos of the forum — in whatever form it took — would be that comments would mostly be kept short. In other words, what you would not tend to do, at least if you wanted to keep within the spirit of things, is spend a month thinking hard about the problem and then come back and write ten pages about it. Rather, you would contribute ideas even if they were undeveloped and/or likely to be wrong.

This suggestion raises several questions immediately. First of all, what would be the advantage of proceeding in this way? My answer is that I don’t know for sure that there would be an advantage. However, I can see the following potential advantages.

(i) Sometimes luck is needed to have the idea that solves a problem. If lots of people think about a problem, then just on probabilistic grounds there is more chance that one of them will have that bit of luck.

(ii) Furthermore, we don’t have to confine ourselves to a purely probabilistic argument: different people know different things, so the knowledge that a large group can bring to bear on a problem is significantly greater than the knowledge that one or two individuals will have. This is not just knowledge of different areas of mathematics, but also the rather harder to describe knowledge of particular little tricks that work well for certain types of subproblem, or the kind of expertise that might enable someone to say, “That idea that you thought was a bit speculative is rather similar to a technique used to solve such-and-such a problem, so it might well have a chance of working,” or “The lemma you suggested trying to prove is known to be false,” and so on—the type of thing that one can take weeks or months to discover if one is working on one’s own.

(iii) Different people have different characteristics when it comes to research. Some like to throw out ideas, others to criticize them, others to work out details, others to re-explain ideas in a different language, others to formulate different but related problems, others to step back from a big muddle of ideas and fashion some more coherent picture out of them, and so on. A hugely collaborative project would make it possible for people to specialize. For example, if you are interested in the problem and like having slightly wild ideas but are less keen on the detailed work of testing those ideas, then you can just suggest the ideas and hope that others will find them interesting enough to test or otherwise respond to.

In short, if a large group of mathematicians could connect their brains efficiently, they could perhaps solve problems very efficiently as well.

The next obvious question is this. Why would anyone agree to share their ideas? Surely we work on problems in order to be able to publish solutions and get credit for them. And what if the big collaboration resulted in a very good idea? Isn’t there a danger that somebody would manage to use the idea to solve the problem and rush to (individual) publication?

Here is where the beauty of blogs, wikis, forums etc. comes in: they are completely public, as is their entire history. To see what effect this might have, imagine that a problem was being solved via comments on a blog post. Suppose that the blog was pretty active and that the post was getting several interesting comments. And suppose that you had an idea that you thought might be a good one. Instead of the usual reaction of being afraid to share it in case someone else beat you to the solution, you would be afraid not to share it in case someone beat you to that particular idea. And if the problem eventually got solved, and published under some pseudonym like Polymath, say, with a footnote linking to the blog and explaining how the problem had been solved, then anybody could go to the blog and look at all the comments. And there they would find your idea and would know precisely what you had contributed. There might be arguments about which ideas had proved to be most important to the solution, but at least all the evidence would be there for everybody to look at.

True, it might be quite hard to say on your CV, “I had an idea that proved essential to Polymath’s solution of the *** problem,” but if you made significant contributions to several collaborative projects of this kind, then you might well start to earn a reputation amongst people who read mathematical blogs, and that is likely to count for something. (Even if it doesn’t count for all that much now, it is likely to become increasingly important.) And it might not be as hard as all that to put it on your CV: you could think of yourself as a joint author, with the added advantage that people could find out exactly what you had contributed.

And what about the person who tries to cut and run when the project is 85% finished? Well, it might happen, but everyone would know that they had done it. The referee of the paper would, one hopes, say, “Erm, should you not credit Polymath for your crucial Lemma 13?” And that would be rather an embarrassing thing to have to do.

Now I don’t believe that this approach to problem solving is likely to be good for everything. For example, it seems highly unlikely that one could persuade lots of people to share good ideas about the Riemann hypothesis. At the other end of the scale, it seems unlikely that anybody would bother to contribute to the solution of a very minor and specialized problem. Nevertheless, I think there is a middle ground that might well be worth exploring, so as an experiment I am going to suggest a problem and see what happens.

I think it is important to do more than just say what the problem is. In order to try to get something started, I shall describe a very preliminary idea I once had for solving a problem that interests me (and several other people) greatly, but that isn’t the holy grail of my area. Like many mathematical ideas, mine runs up against a brick wall fairly quickly. However, like many brick walls, this one doesn’t quite prove that the approach is completely hopeless—just that it definitely needs a new idea.

It may be that somebody will almost instantly be able to persuade me that the idea is completely hopeless. But that would be great—I could stop thinking about it. And if that happens I’ll dig out another idea for a different problem and try that instead.

It’s probably best to keep this post separate from the actual mathematics, so that comments about collaborative problem-solving in general don’t get mixed up with mathematical thoughts about the particular problem I have in mind. So I’ll describe the project in my next post. Actually, make that my next post but one. The next post will say what the problem is and give enough background information about it to make it possible for anybody with a modest knowledge of combinatorics (or more than a modest knowledge) to think about it and understand my preliminary idea. The following post will explain what that preliminary idea is, and where it runs into difficulties. Then it will be over to you, or rather over to us. I’ve already written the background-information post, but will hold it back for a few days in case the responses to this post affect how I decide to do things.

The blog medium is almost certainly not optimal for this purpose, so if a serious discussion starts with lots of worthwhile contributions, then I’ll look into the possibility of migrating it over to some purpose-built site. If anyone has any suggestions for this (apart from the obvious one of using the Tricki — I’m not sure that’s appropriate just yet though) then I’d be delighted to receive them. My feelings at the moment are that blogs are too linear—it would be quite hard to see which comments relate to which, which ones are most worth reading, and so on. A wiki, on the other hand, seems not to be linear enough—it would be quite hard to see what order the comments come in. So my guess is that the ideal forum would probably be a forum: if someone knows an easy way to set up a mathematical forum, I might even do that. But if the discussion is on this blog, then I might from time to time try to assess where it has got to and create new posts if I feel that genuine progress has been made that can be summarized and then built on.

I’ve been thinking of doing this for a long time. The reason I’ve suddenly decided to go ahead is that I followed a couple of links from this post on Michael Nielsen’s blog, and discovered that, unsurprisingly, others have had similar ideas, and some people are already doing research in public. But the idea still seems pretty new, particularly when applied to one single mathematics problem, so I wanted to try it out when it was still fresh. (I would distinguish what I am proposing from what goes on at the n-category café, which is an excellent example of collaborative mathematics, but focused on an entire research programme rather than just one problem.)

To finish, here is a set of ground rules that I hope it will be possible to abide by. At this stage I’m just guessing what will work, so these rules are subject to change. If you can see obvious flaws let me know.

1. The aim will be to produce a proof in a top-down manner. Thus, at least to start with, comments should be short and not too technical: they would be more like feasibility studies of various ideas.

2. Comments should be as easy to understand as is humanly possible. For a truly collaborative project it is not enough to have a good idea: you have to express it in such a way that others can build on it.

3. When you do research, you are more likely to succeed if you try out lots of stupid ideas. Similarly, stupid comments are welcome here. (In the sense in which I am using “stupid”, it means something completely different from “unintelligent”. It just means not fully thought through.)

4. If you can see why somebody else’s comment is stupid, point it out in a polite way. And if someone points out that your comment is stupid, do not take offence: better to have had five stupid ideas than no ideas at all. And if somebody wrongly points out that your idea is stupid, it is even more important not to take offence: just explain gently why their dismissal of your idea is itself stupid.

5. Don’t actually use the word “stupid”, except perhaps of yourself.

6. The ideal outcome would be a solution of the problem with no single individual having to think all that hard. The hard thought would be done by a sort of super-mathematician whose brain is distributed amongst bits of the brains of lots of interlinked people. So try to resist the temptation to go away and think about something and come back with carefully polished thoughts: just give quick reactions to what you read and hope that the conversation will develop in good directions.

7. If you are convinced that you could answer a question, but it would just need a couple of weeks to go away and try a few things out, then still resist the temptation to do that. Instead, explain briefly, but as precisely as you can, why you think it is feasible to answer the question and see if the collective approach gets to the answer more quickly. (The hope is that every big idea can be broken down into a sequence of small ideas. The job of any individual collaborator is to have these small ideas until the big idea becomes obvious — and therefore just a small addition to what has gone before.) Only go off on your own if there is a general consensus that that is what you should do.

8. Similarly, suppose that somebody has an imprecise idea and you think that you can write out a fully precise version. This could be extremely valuable to the project, but don’t rush ahead and do it. First, announce in a comment what you think you can do. If the responses to your comment suggest that others would welcome a fully detailed proof of some substatement, then write a further comment with a fully motivated explanation of what it is you can prove, and give a link to a pdf file that contains the proof.

9. Actual technical work, as described in 8, will mainly be of use if it can be treated as a module. That is, one would ideally like the result to be a short statement that others can use without understanding its proof.

10. Keep the discussion focused. For instance, if the project concerns a particular approach to a particular problem (as it will do at first), and it causes you to think of a completely different approach to that problem, or of a possible way of solving a different problem, then by all means mention this, but don’t disappear down a different track.

11. However, if the different track seems to be particularly fruitful, then it would perhaps be OK to suggest it, and if there is widespread agreement that it would in fact be a good idea to abandon the original project (possibly temporarily) and pursue a new one — a kind of decision that individual mathematicians make all the time — then that is permissible.

12. Suppose the experiment actually results in something publishable. Even if only a very small number of people contribute the lion’s share of the ideas, the paper will still be submitted under a collective pseudonym with a link to the entire online discussion.

228 Responses to “Is massively collaborative mathematics possible?”

1. How do other disciplines solve the problem of giving credit (when it comes time to make decisions for hiring, tenure, etc.) in large collaborations? And for that matter, how did mathematics handle Bourbaki? Of course that is not the same thing, as Bourbaki wasn’t producing new results but rather systematizing old ones.

2. In regard to your #12, let’s say an experiment like this results in something publishable. Whose job is it to “write it up” for publication? (For the record, I dislike the term “writing up”. The preposition “up” somehow trivializes the act, and seems to put a wall between research and writing. This seems a bit silly given that proofs are just pieces of writing constructed with the purpose of convincing people that some mathematical statement is true.) That person will clearly have done some substantial portion of the work; how is this made clear? (A thought is that this person is somehow analogous to the director of a film while people who contributed various results are analogous to the actors, although I suspect this metaphor breaks down if you look at it hard enough.)

In regard to the technical issue of software, I’m almost certain I’ve seen forums which allow:
1. threaded posts, which are of course important so one can tell which posts are replies to which, and
2. inline TeX support, which is important for obvious reasons.
In other words, the software you’re thinking of almost certainly exists.

Some answers off the top of my head. I don’t know what happens with large collaborations in other subjects, but I’ve often wondered how one is supposed to evaluate a chemist, for example, who works as a member of a large team in a laboratory, for hiring purposes.

The writing-up point is a good one. I’d imagine that there would be a discussion about it and someone, or some small group of people, would be chosen or would volunteer. They would continually post links to rough drafts for others to comment on. And, as with the research itself, anybody could look at the resulting conversation and the various drafts and could find out who had done what. Perhaps one could do more than this, though. For instance, once a project was finished, a very brief note could be written to thank those who had done the hard work at the end.

I too am almost sure I’ve seen threaded posts with TeX support. What I’d really like to find, but haven’t found yet, is a place where one can easily and freely set up such a forum. The word “easily” should be interpreted in a very strong sense, I’m afraid, or it will be beyond me.

The Wiki model seems ideally suited to the writing-up phase, if perhaps not to the earlier research phase. At some point Wiki formatting would have to be converted into LaTeX formatting but I think that’s best left as late as possible.

Science is not a public good it is a collegiate good. I have no idea what 95% of mathematicians are talking about only specialists in their area do. People carry out science not to impress and help the public but mainly to impress and help those in their college (in the sense of discipline). The book “sex, science and profits” has a good discussion of how and why science is done.

Another model to incentivise scientific progress is Robin Hanson’s prediction market. For mathematics this would work something like “I bet p!= NP” the market gets odds over time. If i am about to contribute a piece of mathematics that will change those odds i can bet then publish and make a profit. There are some problems with decidability for these issues in maths but Hanson’s idea is an interesting one.

When I once had to present dozen-author paper from a collaborations in computational geometry, I made a slide of the first three pages of a Phys Rev Letters paper about the discovery of the Z0 particle by the “Omega collaboration”, which listed the all 495 authors and their 39 institutions. (The remaining 6 pages were text, tables, and bibliography.)

One delightful feature of the concentrated attention of several people in a room is how person A will completely misinterpret what person B is trying to say, and that will trigger the idea for A to surmount their current barrier.
(Add to your technical requirements: a reader so that people can receive input while their attention is still focused on their own throughts?)

I’ve had quite a similar idea for a while. Say one builds a social network for mathematicians. You could put in ifo about yourself, your CV, papers you like and of course have friends, but the main use would be another.

You could have a “status” where you describe what you have done today, mathematically speaking. This comment could range from “I learnt about this” or “I studied that paper” to “here’s an idea that could work for what I’m doing, I’m testing it next days” or “I finally have a proof of whatever, let me write it here”.

Then the system would save key-words from your posts and relate them to other people posts, in order to be able to automatically suggest friendship. When a friend is suggested, you can see the related posts and decide that his (or her) knowledge is valuable for what you’re doing or viceversa. Then you add him as a friend and propose a traditional collaboration.

In short this wouldn’t replace traditional collaboration, but would help to find people who know exactly what you need.

For idea stealing, the thing goes the same as your proposal. Once I post, my idea is public, so people won’t be able to steal it; rather they can propose a collaboration.

I guess I had some vague idea that this might spur some sort of collaborative attacks on these problems, though I did not explicitly try to organise this, and in any event the problems are all quite difficult (though one of them – scarring for the stadium – has had a major breakthrough recently). But I did get some interesting discussion, for instance the comments on Mahler’s conjecture,

did to some extent resemble the type of collaborative effort you describe here, even if it did peter out eventually.

This type of approach might work best for problems which, while not genuinely parallelisable, can at least generate a number of simpler sub-problems (special cases, ambitious generalisations to be disproven, analogues in other, better understood, branches of mathematics, arguments that make some “cheating” or “best-case scenario” assumptions that eliminate one or more key difficulties, etc.) which can largely be worked on in parallel. If the problem is of the nature that everyone is stuck on initiating Step 1, then it’s not clear that massively multiplying the number of people thinking about that step is the optimal solution.

There is also the issue of keeping the signal-to-noise ratio at a reasonable level; it’s not too difficult to detect and remove obvious nonsense, but one can imagine borderline cases (e.g. one person filling up the forum with attempts at one approach that everyone else has abandoned as a fruitless dead end) that may require some diplomatic moderation, for instance.

This happens to be exactly the way the most well-organized teams solve puzzles in MIT’s annual Mystery Hunt – in that environment, there are a large number of puzzles and a smaller, but still significant, number of people (ideally proportional to the number of puzzles), and people bounce ideas off of each other on a Wiki or something similar until someone has the correct idea and someone else takes it in the correct direction.

(I should mention that while the puzzles are far from easy, they tend to be written to have the property that once you have the right idea it is fairly obvious that you have it, which is perhaps an issue that makes this case sufficiently distinct from the case of mathematics.)

[…] Is massively collaborative mathematics possible? [via nielsen]. One day after Michael Nielsen’s post looking at how blogging creates a new forum for solving scientific problems, Fields Medal-winner Timothy Gowers decides “to suggest a problem and see what happens.” […]

[…] in the remarks.) Michael’s post triggered Tim Gowers to present his thoughts about massive collaboration in mathematics. This is a very interesting post and there are interesting follow-up remarks. Ben Webster in […]

Depending on the actual number of people working on the problem and the ratio of ideas per day they are able to produce, another issue that can arise is how to manage the relation of information: quoting, answering and referencing to other’s ideas can be changelling. Suppose there are 500 commentaries before yours, it wouldn’t be helpful for the rest to say your idea comes from numbers #73, #128, #199 and #383, and it wouldn’t be possible to copy all the relevant information and keep the contributions small, neither would be it desirable, in my opinion.

Some special rules, or even a more structured tool (I don’t think threaded forums are that useful in this case) might prove necessary in order we can track every chain of ideas easily.

Maybe the moderator suggested by Terry Tao could compose summaries (I’m thinking about two kinds: brief summaries and exhaustive summaries) and add them to the discussion (in a parallel tool, not in the main blog/forum) or even replace all the comments with them every once in a while.

This project strongly reminds me some famous words by Evariste Galois: Quand la concurrence c’est-à-dire l’égoïsme ne règnera plus dans les sciences, quand on s’associera pour étudier, au lieu d’envoyer aux académies des paquets cachetés, on s’empressera de publier ses moindres observations pour peu qu’elles soient nouvelles, et on ajoutera: “je ne sais pas le reste”. (Rough translation: When competition, that is egoism, no longer reigns in the sciences, when people join to study, then instead of sending sealed parcels to academies one will rush to publish any of one’s observations if they are new, and one will add “I do not know the rest”.)

On a practical level, moderation would indeed be crucial, and I second Jose Brox’s idea of hiding some of the comments by a summary (e.g. statement of a now proven lemma) as soon as that part of the argument is settled.

Jose Brox’s idea of hiding old comments and replacing them with some sort of summary reminds me of what actually happens on Wikipedia. Wikipedia’s “talk pages” are incredibly messy, but the finished product that most people see emerges from them.

Also, why should only the moderator be the one to compose summaries? When starting on a project like this the moderator won’t necessarily know in what direction things will go, and it’s possible that the solution to a problem will go through areas with which the moderator is not that familiar. Anybody should be able to write summaries. (On a smaller scale, this is like how anybody can write a review of the literature in some area, although of course the people writing review articles are hopefully people fairly knowledgeable about the area they’re writing about.)

Finally, the hypothetical software that we’re talking about could even be useful for a single mathematician, or a more “traditional” (i. e. smaller) team of collaborators. I know that despite my best efforts, I often lose track of the interrelationships between my ideas, which makes things hard when I try to write them up. If I were entering all my ideas into a computer, with the relationships between them indicated (things like “X is a generalization of Y”, “X `looks like’ Y in a different domain”, etc.) it would perhaps be easier to combine them when I write things up. Perhaps this is on my mind because I am working on generating the mathematical ideas that will hopefully go into my PhD thesis; my current main project is thus by a substantial margin the largest piece of mathematics I have ever done, and I’m finding that my old organizational systems don’t scale well.

This type of approach might work best for problems which, while not genuinely parallelisable, can at least generate a number of simpler sub-problem

I might also add that if the problem is too technically formidable, it is likely the discussion will wind down to only 2 or 3 participants (in which case the effort is no longer “massive” but a traditional collaberation).

For instance, this problem Timothy posted about I believe could succumb to a massive effort, because a.) it’s simple in explanation b.) a undergraduate-level approach might work c.) it has simpler cases d.) brute force or luck may help e.) a multi-discipline approach seems helpful (both computer science and statistical methods could contribute, in addition to pure maths) and f.) even an interested amateur could figure out something to do; maybe go back and illustrate previous results with a better graphic layout.

Andrea, your idea certainly sounds interesting. In general there seem to be many ways that the internet could potentially help us pool our resources and for the mathematical community to become more than the sum of its parts. My guess is that, just as people of my generation say things like, “I can’t believe that when I first wrote papers I actually put them in an envelope and posted them to journals!” people in 15 years time will look back with a sort of nostalgic wonder on how we do things today.

Terry, I completely agree that some problems are going to be more suitable for this than others. I’ve tried to choose one that I hope will work, but as I said in my post it is very much an experiment, and if it fails (as the majority of my individual research projects do) I’ll try to learn what lessons I can from the failure and have another go with something else, or perhaps with a slightly different way of organizing the collective attack.

Qiaochu, I’m interested and encouraged to hear that there is a precedent for this kind of thing, and that it has sometimes worked. It’s not obvious to me that the different nature of a mathematical research problem will make it less likely to work for that.

Jose, Tom and Michael, I had thought about people composing summaries. I agree with Michael that one should not make any assumptions in advance about who would do that, but one thing that would almost certainly be a good idea would be to have the summaries in a separate place. A lot will depend on how many comments there are. If there are hundreds, then there is a serious organizational problem, but I don’t expect that. If there are just a handful, then it’s not much of a problem just to read them all. If the number is largish but not too large, then one possibility might be to have a convention that every so often whoever wants to can write a new “initial” post that takes into account everything that’s been learned about the problem, and the discussion starts all over again. (And others could comment on that post and it could be edited in the light of their comments. This is assuming a blog format for the time being.)

Something else I might do if the number of comments is not too large is forward linking. In other words, if someone writing comment 17 says, “This is a response to comment 9” then I could edit comment 9 by putting at the end of it, “There is a response to this comment in comment 17.” It would be a bit of a Heath Robinson approach, but might be quite useful.

Tom, I was very interested by your Galois quotation, which I didn’t know.

Terry, that was also interesting to see, though I’m hoping that in this case the discussion won’t go from zero to a complete solution of the problem in one single step!

Jason, I hope very much that the problem I’ve chosen will not be too technically formidable. On the other hand, it will not be amenable to an undergraduate-level approach: “massive” may be an overstatement, but I hope that there is at least a possibility that many more than three people will make genuine contributions.

Related to the “signal-to-noise ratio” and “managing information” technical challenges, I thought I’d point you to a self moderating forum-like site growing in the programming community right now (www.stackoverflow.com). The relevance to this discussion is how individual collaborators can upvote or downvote contributed material so that both the quality and organization of ideas for a given question is self sorting.

For a distributed collaboration, I agree with the comments here that it would be fruitful to avoid single moderator requirements, as those could bottleneck the process into being serial instead of massively parallel. In as many aspects as possible, it seems the technical solution should adhere to a “power to the people” philosophy.

Also perhaps relevant is that stackoverflow has a reputation system which measures the extent of individual contributions. Earning more reputation gives more moderation powers, but something like that could also be used at publishing time to sort contributions. Their software isn’t available as far as I know (not that they couldn’t be asked about the possibility), and doesn’t support LaTeX, but I thought the ideas could be useful nonetheless.

In an ideal scenario, having a more sophisticated self organizing setup here, one that allowed collaborators to create and vote on associations verses linear up or down voting, would be fantastic. Something like that would truly be in the spirit of your super-mathematician.

It seems like a massive coincidence that only two weeks ago I tried to start a software project aimed at creating just the sort of forum that is called for.

The most crucial element in such a forum is the need for feedback about what’s working and what isn’t. At a basic level, we’re going to need a way to gently let people know that they need to: { be more concise, be nicer, stay on-topic, acknowledge credit }, but I can imagine the need for a whole range of such interactions.

I don’t think the ‘stackoverflow’ model of trustworthiness is adequate, because it doesn’t make much sense to model trustworthiness on a single scale. Some people are going to have good mathematical ideas; others are going to be better about communicating; others will be better at social interaction; and so forth. The basic point is that trust that a person does ‘X’ well is only loosely correlated with their ability to do ‘Y’ well. In order for a large-scale community to function effectively, I think we’re going to need to find ways of expressing that even though you’ve earned quite a lot of trust by doing ‘X’, ‘Y’, and ‘Z’ well, there’s a pattern that your performance at ‘W’ could be better. For obvious reasons, much of this meta-discussion should be kept private or at least semi-private.

So — I’m not sure whether or not I’ll be able to contribute to the mathematics (my training is mostly in hyperbolic geometry), but I have a definite and independent interest in being an important part of the effort to design and create the forum that seems to be called for.

I’ve been thinking for some time about the conditions which could facilitate massively collaborative thinking, and I have a strong suspicion that the answer to the ‘is it possible’ question is a definite yes.

I don’t want to overwhelm the discussion by elaborating on too many of my ideas now, but I would like to mention one: it would be useful to be able to edit a previously-made comment to reflect post-facto thoughts.

Most mathematicians I know tend to err on the side of speaking too infrequently to allow such a collaborative process work well. My recent tendency is in the other direction (as the suggested by the timing of this comment).

In response to your first question, an interesting model is the high-energy physics community, which often has papers with hundreds of authors. My understanding is that such projects typically have archived internal conversation in the form of “Technical Reports”, often authored by just a few project members. This ensures that everyone has a rough idea of who has contributed what to the project, and a way of assessing people’s individual contributions when it comes time to write letters of recommendation, and so on. Take all this with a grain of salt – I’m not an experimental high energy physicist, it’s just what I’ve been told by people who have worked in that community.

Nathaniel, I have the same sense that mathematicians tend to err on the side of speaking too infrequently. We seem to act as if everything we say must be correct the first time anyone else ever sees it.

1. That we use the same code of conduct for this discussion as for the super-mathematician.

2. That we treat suggested rules as in force, unless we have good reason to break them. In particular, I really like the suggestions put forth by gowers, and wish to encourage others to read them carefully and follow them.

3. That someone (or a small focused group) start work on summarizing the discussion so far. I’m willing to forge ahead on this task if a general consensus emerges that I should do so.

4. That we refer to the discussion about the social interaction between the people involved in the “super-mathematician” or similar constructs (i.e., this discussion) as “Social Engineering”. I’m aware of the negative connotations, but think the term so highly descriptive that it’s worth ignoring those connotations and reclaiming the term for our use.

Nathaniel, I had already planned to take the following steps, which go some way towards fulfilling your suggestions. First of all, I was going to modify very slightly the rules I suggested (without changing their essence except slightly in one case) and add one or two more. And I was going to put the new set of rules into a new post, together with some remarks about how I hope things will proceed. Secondly, I planned to have a separate thread available for comments on how the project is going, suggestions for procedural changes that might improve it, and so on: as I wrote earlier, I don’t want such comments to be mixed in with the mathematical ones (though they might well refer to them). I agree that it could make very good sense to summarize that discussion from time to time.

Michel, I have never forgotten a conversation I once had with a mathematician at Princeton, who said, “If someone proved the Riemann hypothesis, there would be a lot of grey faces at the Institute.” The implication was that that particular problem is one that seems to encourage at least some people to do a lot of private work on it. And such people would I think be reluctant to give away their ideas (though as I wrote in my original post, it could instead be regarded as a way of laying claim to those ideas). Another problem is that a lot of people have very detailed technical knowledge related to RH. But perhaps that’s unimportant: it would just mean that the particular community of people able to contribute to that project would not include me. In theory, I suppose, if a big group worked on the Riemann hypothesis and was seen to be serious, then people who might initially have been reluctant to contribute would come on board.

There’s also the amusing practical problem of how to divide up the million dollars if the theorem actually gets proved this way …

But the main reason for timidity is that it doesn’t seem a good idea to try this out on a big problem until one has some reason to believe that it could work at all.

what might be a reasonable approach would be to host the discussion in a “for this purpose” moderated google group so that we’d have the threading in the discussion, and have one of the moderators lift remarks that seem mathematically useful from the ascii form of into an expanding listing of the useful ideas eg on a purpose built page on eg this blog for reference

I often have difficulty saying things concisely because I take too much care to be precise. I think this is a very common pattern among mathematicians; and I think that it is a pattern we’ll have to adjust if we want to get the super-mathematician operating efficiently. To state it by analogy, “bounding boxes are useful”.

Detail follows (if you grok the above paragraph, it’s optional).

Basically, I think of something that’s initially simple, and try to write it down. In the act of writing it down, I think of a way that it’s wrong, and so modify the original sentiment. Then I try to write that part down, and find another error. Before I know it, I’ve spent an hour or so just trying to communicate something that should be really simple.

What’s called for, I think, is a conscious decision to “just say something” which approximates the truth, bypassing the mathematical instinct to get it “just right” the first time. To preserve our credibility and to alert others to the looseness, we can then mark it as an “approximate statement”.

This is also relevant for preparing summaries. I think gowers’ post on the math background required for contribution is a good example — he begins with an summary which would allow people to skip the rest, and then goes on with the details. Similarly, when summarizing a complex discussion, I think it’s a good idea to deliberately overstate the hypothetical propositions being discussed, since often it will be the case that even the overstated proposition isn’t relevant to what someone else is doing.

Tim Gowers asks, “Is massively collaborative mathematics possible?”
The answer is obviously “yes,” since he also gives an algorithm to implement massive collaboration. Another question might be: Should one do it? and the answer is: Do it if you want to (it might be fun), but don’t do it if you don’t want to. A third question: Is massively collaborative mathematics good or bad? and the answer is: Neither, since there’s no moral or ethical issue here.

There is a fourth question: What do you think about massively collaborative mathematics? I think it’s horrible. My view is based on the romantic (admittedly, nineteenth century, while here we are in the 21st) notion that science and art is produced by the solitary artist (physicist, mathematician, poet, playwright, et al.) attempting to create work that is great in both content and craftsmanship. I suspect that even in the twentieth century, almost all of the new ideas in mathematics originated in papers written by a single author. For example, reviewing the number theory papers of the great collaborator Paul Erdos, I find that in his early years, from his first published work in 1929 through 1945, most (60 percent) of his 112 papers were singly authored, and that most (not all — we can all cite a few counterexamples) of his stunning papers in number theory were papers that he wrote by himself. A glance at MathSciNet today shows that only two of Tim’s 42 papers have a co-author.

Massive collaboration can certainly produce useful results. Tim cites the classification of the finite simple groups as a kind of massive collaboration, and this is a perfect example. It is useful. Ignoring the contentious question of whether the proof is correct (though this is an inherent problem of the collaborative process), the work is definitely boring. As far as I know, neither brilliant insights nor new techniques have come out of the proof and been applied to create new areas of mathematics or solve old problems in unrelated fields. It is more engineering than art. Recalling Kac’s famous division of mathematical geniuses into two classes, ordinary geniuses and magicians, one can imagine that massive collaboration might produce ordinary work or even work of ordinary genius, but not magic. Work of ordinary genius is not a minor accomplishment, but magic is better.

[…] polymath1 | by Terence Tao My good friend Tim Gowers has just started an experimental “massively collaborative mathematical project” over at his blog. The project is entitled “A combinatorial approach to density […]

I can’t speak for others, but as for my own research, at least half of my papers are joint with one or more authors, and amongst those papers that I consider among my best work, they are virtually all joint.

Of course, each mathematician has his or her own unique research style, and this diversity is a very healthy thing for mathematics as a whole. But I think 21st century mathematics differs from 19th and early 20th century mathematics in at least two important respects. Firstly, the advent of modern communication technologies, most notably the internet, has made it significantly easier to collaborate with other mathematicians who are not at the same physical location. (Most of my collaborations, for instance, would be non-existent, or at least significantly less productive, without the internet.) One can imagine the next generation of technologies having an even stronger impact in this direction (with this project possibly being an example; other extant examples include Wikipedia and the Online Encyclopedia of Integer Sequences).

Secondly, the main focus of mathematical activity has shifted significantly towards interdisciplinary work spanning several fields of mathematics, as opposed to specialist work which requires deep knowledge of just one field of mathematics, and for such problems it is more advantageous to have more than one mathematician working on the problem. (Admittedly, much of 19th century mathematics was similarly interdisciplinary, but mathematics had a much smaller diameter back then, and it was possible for a good mathematician to master the state of the art in several subfields simultaneously. This is significantly more difficult to do nowadays.)

The largest collaboration I have been in to date has involved five people – but already the dynamics of research change dramatically at that scale (especially when all five people are in the same room at once). One can toss an idea out there and have it debated by two other collaborators, while a fourth makes comments and corrections from the sidelines, and a fifth takes notes. Connections are made much faster, errors are detected quicker, and thoughts are clarified much more efficiently (often, I find one of my collaborators acting as a “translator” to distill an excited inspiration of another). It may not be “magic”, but it is certainly productive, and actually quite a lot of fun.

[…] Tim Gowers is doing something quite interesting with his blog, starting with a post entitled Is Massively Collaborative Mathematics Possible?, he’s proceeded with a bit of background and a problem to be solved, after which he started […]

By the way, one significant cultural inhibitor to having contributions (particularly the crazy “blue sky” contributions) to these sorts of projects by professional mathematicians is that, as a whole, we are quite reluctant to say anything on the public record which may end up being wrong, foolish, or naive, lest this damage a hard-earned mathematical reputation. (More generally, this seems to be a major inhibitor against transparency in just about any sphere of human endeavour.) I would imagine, though, that should these sorts of things end up being successful, the culture may shift – the level of professional care and caution which is suitable for a published journal is certainly not the same as the level which would be optimal for this much looser forum.

I don’t think it would be a large overstatement to say that we already have, here in this forum, a nascent “massively collaborative meta-mathematician”.

I’m in the process of trying to start a “collaborative programmer”, for the purpose of designing and building a more suitable forum for this “collaborative mathematician” and other “collaborative thinkers”.

Specifically, I am now drafting an introductory document and setting up the initial forum, but until I finish that, any interested parties are invited to contact me, and I’ll be happy to share the collection of notes I have so far.

Regarding the “division of mathematical geniuses into two classes, ordinary geniuses and magicians”:

I have no difficulty in imagining rules that would allow the presence of a single mathematical genius of the magical kind to imbue the entire collaborative mathematician with the benefits of his or her “magical” abilities.

This is a comment about assigning credit in a field where there are massive collaborations. What often happens in practice is that recommendations (oral or written) become a lot more important. The way a serf—er, I mean, a grad student—gets a job is that the nobles—I mean professors—talk to each other and figure out who gets what job. Of course this is what happens even in “individualistic” fields like mathematics, but the political aspect assumes much greater importance in fields where you can’t assess how strong someone is just by looking at publications but must work with them for a while, or ask someone who has worked with them for a while.

When politics becomes this important, it becomes hard for someone who doesn’t come from a famous school, or who simply didn’t get along with his or her advisor, to break through the “old boy network” on merit alone.

I don’t think that the fact that a massive Wiki collaboration stores all the information in a publicly accessible database helps as much as you might think at first. Who has the time or inclination to dig through such a chaotic mass of information to extract the Truth? Besides, there’s always the chance that something that seems clear from an examination of the Wiki will in fact be seen to be way off the mark by someone who knows the full story of what happened, including offline events that were not recorded. The glut of information on a Wiki may in fact drive people making hiring decisions to rely *more* on the grapevine than on the written record.

By the way, I don’t know if anyone has mentioned “The Bourbaki Gambit” by Carl Djerassi. It is a fictional work but it is a fascinating exploration of some of the benefits as well as the pitfalls associated with collaborating under a pseudonym. Djerassi is both an excellent scientist and an excellent writer.

Usually, in mathematics, the authors order of the paper is alphabetic. I do not understand it. However, I do not think that is a good way.I have talked with one of the authors of some papers. I find that some author do not really understand their paper, even he is the first author. Sometime, he will say “you can ask *****”. Unfortunately, I always meet this case. I even know somebody who have big prize but she does not really know her paper for this prize, which makes me confuse. Can I trust paper? Can I trust prize? What can I trust???? I think that it is good way for changing the author orders, which is only related with the contribution to the paper. Or, make a comments to show the contribution of every author.

A small remark about the previous comment: there are numerous examples of singly authored papers where the author (after a suitable time lag) doesn’t really understand the paper. There’s even a famous example (the details of which I’m afraid I’ve forgotten) of a result that was independently proved by three people of whom two were different time slices of the same person. So one should be very careful about drawing any conclusions from conversations of the kind you describe.

Thanks for Gowers’s comments. I really understand that the author may forget his paper after a long time. However, I talk with some authors just for the very recently papers……… Maybe, the case I meeting is a little special. Anyway, as a begginer, I do not understand why the authors order is alphabetic in mathematics.

There are several reasons (that I can think of) for the convention of listing the
authors alphabetically, all of which I think have a lot of merit. Here are two:

(1) When approaching a mathematical problem, it is often not clear at the beginning
which approach will work and which will not, or which subproblems will ultimately
be important and which will be tangential. So the division of labour in a collaboration,
and the amount of work that collaborators put into their part of the joint project,
will not always be reflected in their respective contributions to the final paper. (It
might be that one collaborator works very hard on an approach which later turns
out not to work, or to be subsumed by a different approach.)

There are lots of variants on this point, all of which go to explaining why the
division of effort in the collaboration may not always precisely match the division
of contributions to the final paper.

Since it is hard to predict how such things will go (it is often just a matter
of luck), an alphabetical listing of authors is simpler and fairer than an
after-the-fact analysis of who contributed what to the final paper, and in
what proportion.

(2) Often in collaborations, people at different levels in the academic
hierarchy are working together. I’ve always found it pleasantly democratic
that the authors’ names will simply be listed alphabetically, rather than
(e.g.) the senior author’s name always coming first. (My understanding
is that this is not the case in all other fields, and I find the convention in
mathematics fairer, and more protective of beginning researchers.)

Also, one should bear in mind that in a collaboration in which authors
have different areas of expertise, it is not unreasonable for one author
to defer to the expertise of their collaborator in the area in which the
collaborator works. After all, one reason for collaborating is to bring
together people with different areas of expertise.

Finally, echoing Tim’s remark above, I would hesitate to draw conclusions
about a person’s understanding of their own (or other’s) papers from
casual comments. The level of focus and understanding that someone
can bring to the internal dialogue of their own mind, or to their dialogue
with a close collaborator, is frequently of a totally different order
of magnitude to the one they might bring to a casual conversation in the
tea room.

Some mathematician’s are always “on”, and ready and able to explain
their (or other’s) work at any and all times, but I would guess that those
individuals are more the exception than the rule.

This is a great idea! I was trying to find something like this for a long time.

Do you think CC-By +/- SA would do as a license for this work?
Someone asked who would be in charge of publishing the results. My answer would be another question: Who is in charge of publishing hard cover books of Wikipedia articles?

Delurking just to add to Tim and Matthew’s remarks: explaining what you’ve done is a lot harder than doing it, in my personal view. This is why teachers have to learn, whether by themselves or institutionally, how to teach, not just what to teach.

Oh, and I think that as an accepted convention across a broad spectrum of mathematics articles, alphabetical order of authors is excellent. My name appears at the front of all joint work to date, and it’s good to know that the intended audience will attach precisely *no* weight to that fact whatsoever 😉

One thing “large collaborations” can be very good for is big foundational projects, in the right conditions. I’m not thinking of Bourbaki here (since the rate of appearance of the Bourbaki volumes was never very fast, and the number of collaborators not that large at any given time) but of the rewriting of algebraic geometry by the Grothendieck school. Although the individual pieces typically appeared under single authorship (with the Dieudonné-Grothendieck EGA’s being almost an exception), it really seems that the whole series of SGA volumes, together with the massive theses of people like Illusie, Raynaud, Berthelot, Giraud, etc, amount together to a very massive and well-defined project directed by Grothendieck where the amount of discussion and active collaboration was remarkable. (This being before internet and such things, it was very much localized in space, however).

It’s actually not clear to me however if suitable circumstances could arise again to make a similar project workable today…

“In mathematics we agree that clear thinking is very important, but fuzzy thinking is just as important as clear thinking.”

— Harish-Chandra

“Discovery is the privilege of the child: the child who has no fear of being once again wrong, of looking like an idiot, of not being serious, of not doing things like everyone else.”

— Alexander Grothendieck

“Without fantasy one would never become a mathematician.”

— Marius Sophus Lie

“When a person works, he must have knowledge or he will make terrible mistakes. But at the same time, knowledge alone doesn’t do anything new. You must have instinct and somehow be conscious of making use of instinct. It is an interesting question how to give kids knowledge without having them lose their instinctive power. If you just keep pounding them with knowledge, most lose their instinct and try to depend on knowledge.”

— Heisuke Hironaka

“One of the reasons we don’t do as well as we should is that we are all over-taught.”

— Israel Moiseevich Gelfand

Why are mathematicians so reluctant to say anything on the public record which may end up being wrong, foolish, or naive? Is it really just their fear that the practice might damage their professional reputations? If so, then it would seem they must be shockingly ignorant of the history of mathematics. For, again and again, the most successful masters of the art, from Euler to Atiyah, have spoken eloquently of the need to swashbuckle — and they have practiced what they preached, too.

Just to drive this point home, let me quote a few more examples:

“To discover something in mathematics is to overcome an inhibition and a tradition. You cannot move forward if you are not subversive.”

— Laurent Schwartz

“Undoubtedly, the capstone of every mathematical theory is a convincing proof of all its assertions. Undoubtedly, mathematics inculpates itself when it foregoes convincing proofs. But the mystery of brilliant productivity will always be the posing of new questions, the anticipation of new theorems that make accessible valuable results and connections. Without the creation of new viewpoints, without the statement of new aims, mathematics would soon exhaust itself in the rigor of its logical proofs and begin to stagnate as its substance vanishes. Thus, in a sense, mathematics has been most advanced by those who have distinguished themselves by intuition than by rigorous proofs.”

— Christian Felix Klein

“What gave me a place among the mathematicians of our day, despite my lack of knowledge and form, was the audacity of my thinking.”

— Marius Sophus Lie

“If mathematics is to rejuvenate itself and break exciting new ground it will have to allow for the exploration of new ideas and techniques which, in their creative phase, are likely to be as dubious as in some of the great eras of the past. Perhaps we have high standards of proof to aim at but, in the early stages of new developments, we must be prepared to act in more buccaneering style.”

— Michael Atiyah

It seems to me far-fetched to suggest that most mathematicians would quarrel with any of these views. Everyone, I think, knows perfectly well that Euler, Riemann, and Poincare advanced our art tremendously, despite their failure to prove (as we now understand the meaning of that word) a significant fraction of their most influential results.

Moreover, I doubt very much that any of the following observations will seem at all controversial to the readers of this blog:

“I think that a felicitous but unproved conjecture may be of much more consequence for mathematics than the proof of many a respectable theorem.”

— Atle Selberg

“In the work of the scientist, formulating the problem may be the better part of the discovery, the solution often needs less insight and originality than the formulation.”

— George Polya

“In fact, mathematics is not just about proving theorems, but it’s about finding imaginative and productive ways to think about broad areas of abstract reason.”

— Curtis McMullen

“Very often in mathematics the crucial problem is to recognize and discover what are the relevant concepts; once this is accomplished the job may be more than half done.”

— Israel Nathan Herstein

So: with all this “cover” extended to them by the some of the most eminent masters of the art in all its long, rich history, why do so many mathematicians still feel so uneasy about thinking fuzzily in public? Why is “blue sky” thinking still considered somehow disreputable? If Sophus Lie attributed his own success as a mathematician to the audacity of his thinking, then why are we all so timid?

Of course it’s true that some of us — notably, by the way, the proprietor of this very blog, who recently observed that

“Mistakes are an important part of the thought process. It’s a good research strategy to be a little bit cavalier about the details and go back and check them afterwards. It speeds up the process of having ideas.”

do in fact continue to stress, publicly and vigorously, the need for a certain wildness of mind in pursuing the art of mathematical investigation.

But the clear majority of mathematicians still seem very strongly disinclined to let their hair down and swashbuckle in the grand old manner, especially when they know that others can see them. Despite the urgings and object lessons of their many illustrious forebears, and despite the indisputable value (glory, even) that attaches to success in the project, they just can’t seem to let themselves go.

Part of the problem, I think, is that swashbuckling is an art, and we don’t really teach it. Many mathematicians hesitate to swashbuckle in public for the same reasons that they hesitate to dance in public. But this really just begs the question. For if we (as a community, anyway) know how to swashbuckle, and if we agree that swashbuckling is a crucially important skill, then the real question is: why don’t we teach it?

And here I think the answer is that many of the greatest swashbucklers are notoriously inarticulate; their knowledge is what Michael Polanyi would call “tacit.” It is like the knowledge that native speakers of a language have of its proper use. Though tremendously rich, various, and reliable, this knowledge does not take the form of detailed linguistic rules, arranged in a clear logical hierarchy, and recasting it into that form is a vast and daunting project.

(I invite native speakers of English to contemplate composing, based on their own acquaintance with the language, something like Quirk and Greenbaum’s COMPREHENSIVE GRAMMAR OF THE ENGLISH LANGUAGE — a remarkable, and even a humbling, work.)

This analogy to the tacit knowledge of native speakers, by the way, seems to appeal also to William Thurston, who has noted that

“Studying mathematics one rule at a time is like studying a language by first memorizing the vocabulary and the detailed linguistic rules, then building phrases and sentences, and only afterwards learning to read, write, and converse. Native speakers of a language are not aware of the linguistic rules: they assimilate the language by focusing on a higher level, and absorbing the rules and patterns subconsciously. The rules and patterns are much harder for people to learn explicitly than is the language itself. In fact, the tremendous and so far unsuccessful attempts to teach languages to computers demonstrate that nobody can yet describe a language adequately by precise rules.”

And this, I think, gets us pretty close to the root of the problem.

For anyone who has studied a foreign language seriously knows that although the first year or two may largely be given over to assimilating explicitly understood patterns (regular verb conjugations, rules for the use of prepositions, etc.) it really isn’t long before much subtler things start to happen. And in the end, one learns to do many things (correctly!) for which one simply cannot offer an explanation. The crucial element that makes this possible is immersion in the language, because the native speakers from whom one learns these things do not (indeed, cannot) teach them by explaining them.

On 22 November 1888, Hermite wrote to Mittag-Leffler to express his frustration with Poincare. “Poincare shows the way and gives the signs,” said Hermite, “but leaves much to be done to fill the gaps and complete his work. Picard has often asked him for enlightenment and explanations on very important points in his articles in the Comptes Rendus, without being able to obtain anything except the statement: ‘It is so, it is like that,’ so that he seems like a seer to whom truths appear in a bright light, but mostly to himself alone.”

The trouble here is that Poincare was the native speaker of a language that Picard, Hermite, and Mittag-Leffler had to struggle even to parse. Poincare’s knowledge wasn’t mystical; it was just beyond his powers to articulate. And, as simple examples show, there’s nothing especially strange about that. Native speakers of English will likely all agree that of the two sentences

(1) Did you notice the funny way Bob looked at Alice?

(2) Did you notice a funny way Bob looked at Alice?

the second is somehow “wrong.” But why, exactly, is it wrong? What’s the problem with it? And, most of all, how would you explain what’s wrong with it to a native speaker of Japanese?

This, I’m afraid, is exactly the sort of question with which our best swashbucklers are constantly confronted, and it ought to give us a little more sympathy for Poincare’s predicament than Hermite could muster. For, even if they were hard for others to follow, wouldn’t you rather spend your time writing sonnets than composing a tome like Quirk and Greenbaum? If asked to explain why enough sentences like (2) above are wrong, wouldn’t you eventually just answer, as Poincare evidently did, “it is so, it is like that,” and go back to your poetry?

I think, in the end, we’re just reluctant to admit that a great deal of the knowledge that matters most to us is inarticulate — that we can sometimes be sure of something, and be correct, without really knowing why. But the truth is a stubborn thing, and the truth is that people often know more than they can say — in fact, that’s exactly why they’re so valuable. People like Poincare may not be mystic seers — their tacit knowledge is, I’m sure, the result of immersion rather than divine inspiration — but they are extraordinarily hard to replace, or to duplicate.

Can we hope to manage without them, though, if we bring to bear on our problems a vast collective intelligence? Might we achieve through massively collaborative investigations results as wildly original and imaginative as those of the best natural swashbucklers? It would certainly be nice if we could, and I think there are some genuinely exciting reasons for optimism about this possibility.

But it may be prudent to leave to Georg Frobenius, a notoriously sharp-tongued critic, the last, skeptical word:

“Organization is of the utmost importance in military affairs, as it is…for other disciplines where the gathering process of practical knowledge exceeds the strength of any individual. In mathematics, however, organizing talent plays a most subordinate role. Here weight is carried only by the individual. The slightest idea of a Riemann or a Weierstrass is worth more than all organizational endeavors.”

This is an interesting question. I think there is quite a bit to be learned from the experience of open source development, as described for example in a very nice book “The Success of Open Source” by Steven Weber. One gets the sense that some things are possible and others are not, and that to make such projects work one needs a serious organizational structure behind the scenes.

I didn’t have time to digest the long discussion so maybe this point has been made, but a good example of a hard problem which was solved by explicitly collaborative mathematics might be the classification of the finite simple groups. Rather than base the discussion on particular new media or technologies (wikis, blogs, etc.), one might be better off looking at successful past efforts and then developing technology or media to support the styles of interaction which seemed to work.

It could be that this attempt at “massively collaborative mathematics” actually reaches at a much, much bigger question than just massive collaboration, by exposing the “dirty work” it is likely to shed some light on the roots of creativity in mathematics, a topic which hasn’t been much delt with since the works of Poincaré and, Hadamard in The Psychology of Invention in the Mathematical Field.

The emphasis seems to be on solving problems. The 1974 McLone report on mathematics graduates in employment wrote: “Good at solving problems, not so good at formulating them, the graduate has a reasonable knowledge of mathematical literature and technique; he has some ingenuity and is capable of seeking out further knowledge.”

A practical concern is, who gets there name on the paper? Everyone that contributed to the “brain storming”? Or just the “major contributors”? Both have problems. For the former, that people who didn’t actually contribute would get credit while not producing anything, and the latter is basically, where do you draw the line of how much input or quality thereof gets credit? Undoubtedly, there will be people feeling that they got “shafted.” That likely leading to many people not participating in future massively collaborative events and not speaking fondly of it. In other words, this structure is likely to fail if only because of the human factor.

Another issue would that people, especially Profs, don’t like looking “stupid.” So, they’ll be quite unlikely to contribute something that might turn out to be that way. Not to mention that the training that Mathematicians get is completely opposite to just posting an idea without giving it much thought. So, in general, participation would be questionable.

Clearly, there are other “human” issues as well.

When it comes to the method itself, there is also the problem of the deluge of nonsense. As in, if successful, the amount of input from a truly massive collaboration would be *a lot* to sift through. It’d also make it likely that many viable ideas will “fall through the cracks.” This will be exacerbated if the forum for this discussion would be open to the general public (if it isn’t, how is it decided who gets access or even knowledge that the discussion exists?). Such things tend to attract the “crackpots.” So, there will be a *very* low signal/noise ratio.

But, all in all, I really don’t think that this is a good idea if only because of the above. That isn’t to say that it can’t work on a higher level (i.e. not problem specific). But, the more specific the problem gets, the less likely things such as this will turn out well. There are probably relatively very few authors on Maths papers for a reason.

Addendum: I find that publishing “under a collective pseudonym”, rather impotent. What I mean by that, is that although for those already established it doesn’t really matter, for those that are trying to build a reputation, how does being involved in a discussion that resulted in a publication really increase reputation when the actual contribution is buried in a massive conversation? At least when ones name appears on a paper there is a good guarantee that the person made a significant contribution. But, with this sort of thing there really isn’t. And who is going to go through these threads to determine how much the person contributed to the idea? How do you track overall contribution when the pseudonym would constantly change? It’s rather intractable and it makes the number of and significance of the publication moot when it comes to seeing it on a CV. After all, one could just go around and participate in as many of these conversations as time allows. Then after a time they’ll have gotten several “publications” and could then go around acting like they’ve done something they really haven’t. Or they could just lie. After all, who’s going to be able to check it? It’d be an HR nightmare.

I could see this sort of thing working on a smaller scale. Perhaps for one group. Perhaps a couple. Then when a problem is being discussed, if the conversation diverges, someone could “bank” the idea for later consideration and get things back on track. The result would be a documented conversation between colleagues (even if its just on-line notes) and a list of potential research paths that could be considered once the current one has been completed.

In a larger group, this could be done simultaneously. A subgroup of people initiate research and have some diverging thoughts that are “banked.” Several other subgroups do the same. But, they clearly won’t all end there research at the same time. So, once complete, the subgroup could go back to the “bank” to see what’s there. In an open setting, they’d have access to every ones ideas. So, they might be able to start a “banked” idea while the originator(s) is/are still working on his/her/there research. Of course, subgroups may or may not stay the same.

But, this brings up the credit issue again. What if the person who came up with the idea doesn’t want to pursue it now, but at a later point? What if they don’t want to pursue it at all? How will credit be given to him/her/them?

This idea is interesting. But, there are a number of social issues that need to be overcome before it can be practically implemented. Not to mention figuring out where that line is where such a thing is beneficial.

At any rate, it’s 5am here and I’m a little cross-eyed at the moment. Hopefully, I haven’t come off poorly. Sorry, if I have.

As a last comment today (I promise this time), how will you ensure that the link to the conversation that resulted in the publication remains relevant? In other words, the internet is in a constant state of flux. Links that are good today, may be broken tomorrow. Data is constantly written and erased. These conversations will be no different if only because, at some point, an upgrade in the software used will change things and, at least, alter the links. There’s also HDD crashes and other related failures (I had that happen to me twice last year – the backup failed as well by the way, both times). It’s bound to happen at some point. And then the pseudonym and its related discussion become broken.

The internet may have brought about a massive improvement in speed of communication. But, with it has come a profoundly brittle storage of that information. These concerns need to be addressed if we are to rely on such things for the storage of something as important as Mathematics.

@Laurens Gunnarsen:

One must be careful to not misinterpret those quotes. Sure, it is great to come up with something new and innovative (crackpots routinely use this as a defence of there crackpottery). I doubt many would argue that. But, there is a flip-side to that coin. Many a career has been ruined by prematurely releasing resulted that were ultimately incorrect. It is (typically) only those that are *already* well established that are able to make such leaps and fail, yet still survive. After all, if one is routinely wrong while trying to establish a reputation, ones reputation will be that (s)he is routinely wrong.

With respect, kids are the last people that this should be tested on. After all, the complexity of the problem “we” are talking about is *well* beyond there capability. And since with an increase in complexity, the dynamic of how one needs to work with others changes (not to mention the problem itself), any simple task that the kids might be given wouldn’t yield any usable information about the productivity of this method. At best it’d produce a false positive/negative.

Regarding ‘open sourcing’ Mathematical solutions, check out http://arxiv.org/ and other pre-print servers. In other words, that’s already happened (to at least a large degree). Not to mention that the usage of results is unrestricted (with citation) and anyone that is actually researching will have access to the journals through the institution they work for (e.g. there University). This issue is also getting better, not worse. There have been several cases of researchers releasing there work under a creative commons type license and the journals getting in trouble for not obeying the terms of that license. I believe a few of these stories were linked to from slashdot. So, that might be a place to find them aside from Google if you’re interesting in the details.

@reidalthough for those already established it doesn’t really matter, for those that are trying to build a reputation, how does being involved in a discussion that resulted in a publication really increase reputation when the actual contribution is buried in a massive conversation?

Just like open source software developpers “pay off” is quite different from conventional developpers, open source research will certainly entail very different outcomes for the participants than standard academic publishing.
The proof in the pudding, just let the experiments go and don’t worry about “who gets there name on the paper”, “looking stupid”, “viable ideas falling through the cracks”, crackpots, broken links, HDD crashes, etc…
The actual participants of polymath1 are at no risk it seems and the Internet is a well known mess yet it works, while the kind of objections you raise is exactly what prevented the much earlier Project Xanadu to ever take off, Worse is better.

Yah, sure Kevembuangga. Let’s not worry about any practical issues that entail some semblance of reality. Let’s not worry about whether people are actually good researchers and are able to prove that. Let’s not worry about whether any and/or all this research will vanish in the blink of an eye. Let’s not worry about the flood of crackpots that has already caused problems on the arXive and many a persons/researchers blog, etc, etc, etc.

Ignoring valid practical very *very* real problems is delusion.

Point of fact, your example objections are apples and oranges comparisons; at best sophistry. What we are talking about here are people that are capable of doing real Maths research. And those people are so much more rare than programmers it isn’t funny. Open-source is a *very* different beast. I should know, I participate in it. So, to apply such ideas in Maths without giving it any thought, is a *very* bad idea. Not to mention a waste of time and grant money.

Also, worse is better?!?!? Are you serious?!?!? Do you honestly think that the journals will publish worse material? Do you honestly think that it is beneficial to waste time on sifting through crap? Because, that’s the whole point isn’t it. To get to a place that is more efficient that still results in a publication. The journals will /not/ put up with worse. And I certainly don’t see any benefit in any scheme where any crackpot can post his/her crackpottery. Especially, when the choice is either to sift through it (*big* waste of time) or have some sort of moderating system like Slashdot. A system that is inherently grossly flawed.

Because, that kind of system is *very* flawed. Ideas will either rise or fall based on the bias of the people moderating. Ever post something against RMS in a thread about him? Something true? What happens? You get modded into oblivion is what happens. Similarly for all the academic threads. If you post something that is a valid concern, but someone who likes the idea comes across your post, they’ll mod you down. To think that such things won’t happen in the Maths setting is naive beyond belief.

Point of fact, such moderation systems are a crap shoot at best. And most of the time they don’t really work. It’s just that people don’t see what was modded down, or didn’t get modded up because they can’t be bothered to look at the posts at that low of a level. It’s just too much of a time commitment and people are lazy and/or have way too much other stuff to deal with e.g. classes, administrative demands, marking, training TA’s/lab assistants, etc, etc, etc.

As I said, this idea /could/ be beneficial on a smaller scale. But, the “massively” way… not so much. Too many problems in too many areas.

Reid — you have something important to say, and you have said most of it. While I don’t agree with your conclusion that this is necessarily a bad idea, I will give an enthusiastic second to the notion that it should be approached with a great deal of caution, lest it spin out of control, and I’m very much in favor of engaging in a respectful discussion about the risks and rewards of this sort of collaboration.

Perhaps I can attempt a calming comment here. I think that Reid introduces some valid concerns, but it’s important to distinguish between two possible scenarios when one is thinking about open-source mathematics. One is that it gradually takes over to the point where pretty well all mathematics is done that way. Then it really is the case that we would have to rethink in a big way how we assign credit, and at the moment it looks to me as though what we would end up with would be quite a bit less satisfactory than what we have today. (I am not sure about this, but there are serious problems that I don’t see how to solve.) The other scenario is that open research with several collaborators becomes something that some people feel like doing for some problems, not because they want to take over the world, but just because it is fun, and because the resulting conversation, though messy and full of ideas that go nowhere, is interesting by virtue of that very fact: it shows you the things that the journals and textbooks hide.

With these concerns in mind, and also the concern that an open-source project on some topic could accidentally put someone in a very awkward position if they happened to be working on the same topic and had already made progress, I will be choosing future polymath problems quite carefully, with the aim of keeping the whole thing entertaining while not encroaching on the territory of normal, private mathematics.

I’d like to emphasize a small point gowers just made. I think that “because it is fun” is crucial. Whatever other motivations people might have, I’d like to assert that “it is fun” ought to be a requirement for ongoing participation in a collaboration like this.

All of these issues are indeed solvable. Outside of the blogging (e.g. Blogger/WordPress) format, Google has introduced Knol, which I am finding to be invaluable. The company is finally getting more serious when it comes to content hierarchy (i.e., folders and categorized search) in the product, and each knol entry has the option of yielding moderated collaboration. So, for instance, once can start a very specific topic like ‘Advances on the Fefferman-Stein decomposition theorem’ or ‘A new lemma for 3-epps embedded in some space’ on Google Knol, open the collaboration to moderated (limited) accessibility, and go from there. (A total sales pitch, here 😉 Importing Google Documents into the knol is feasible, as is having an online pre-publication video conference. Both tools (realtime document editing + webcam conferencing) are the future, so what better way to experiment with both than through this method (?).

Another issue I’d like to raise pertaining to large-scale activities such as these, is that they are done quite often in astronomy. It’s not uncommon for a paper to have authorship well-into the dozens for a single project; many of whom may or may not be familiar with one another (or their work), but still must face the restrictions of working remotely. Usually, collaborations in this field happen precisely with differences of location being encouraged, since it’s necessary to verify an observation of some astronomical object in different timezones.

I’m seriously giving this problem some thought, but I’ll have to hiatus this for the moment and get some real work done. In the meantime, look into Google Knol.

Regarding your quote “The next obvious question is this. Why would anyone agree to share their ideas? Surely we work on problems in order to be able to publish solutions and get credit for them. And what if the big collaboration resulted in a very good idea? Isn’t there a danger that somebody would manage to use the idea to solve the problem and rush to (individual) publication?” ..

>> I’ve resigned to using Steve Ballmer’s software analogy: You shouldn’t have a computer (PC) and a typewriter both in simultaneous use. Either you are using the digital writer, or the oldschool typewriter, but not both. There needs to be a commitment to go entirely one way or the other; in this case, to e-print. Coming from industry, when I read you academics’ assessments, I’m seeing that you people are having a difficult time pulling away from the old ‘publish-or-perish’ mindset; the same one that says that your name must be published in some hardcopy manual/journal somewhere, or else you don’t matter. I had thought that the arXiv had removed that sense of self-depreciation (G. Perelman STILL hasn’t bothered to print out his Poincare solution or submit it to one of the acclaimed journals you hold in such high esteem), but I guess not.

Blogs are just fine. They are meant to be containers for announcements and such (what I’ll call Tier 1). As long as they are kept as that , then their purpose is more than relevant for these types of projects. The next tier (Tier 2) has to be the actual collaboration client, which I’m pressing researchers use Google Knol for. Instead of opening-up the research knol completely, I say keep it moderated, and anybody with serious credibility should be able to contribute. This level has no real ceiling (or, at least an artificial one). I’d say anyone in/from academia with a graduate degree, or an advanced undergrad, or even advanced high schoolers (e.g. math olympians) can have their say. Those from industry, such as myself, with similar credentials should also be able to contribute.

Once the objective has been met (a.f., a proof is offered and confirmed), then close collaboration and extract a secretary for writing the whole thing up (Tier 3). This is probably the easiest step, since it’s been in the works for some time at this point. Tier 3 is really just a way to formalize the edits. Afterwards, there’s really no need to print it out on paper, just leave it there. If in demand, I’d put it on Google Base, the arXiv, or whatever; but that’s just for possible offline retrieval purposes.

.. Which brings to our attention the issue of fraud, plagiarism, and/or thievery. So, let’s just say some clever grad student (or anyone) is looking to profit from this by sitting back, not contributing – but taking notes – and is attempting to publish the work in some journal without giving credit when the body of work is winding down. Okay. For starters, the turnaround time for journals is notoriously long. While the rest of us are working at the speed of thought, this student’s Tier 3 attempt is ultimately doomed because all notes, sketches, announcements, etc. is right there in fine print for all to see. This is, of course, considering that the student would not provide references or hand-out due credit.

Tier 4 is the substance of our knowledge economy itself. The ONLY time leaf paper should be used here is for some strange reason people can’t get on the Internet and must read the article offline. Then a hardcopy can be provided. But the main points have been made.

Coming from industry, when I read you academics’ assessments, I’m seeing that you people are having a difficult time pulling away from the old ‘publish-or-perish’ mindset;

I’m sure this was intended as a gentle observation, LS, but — as a middling postdoc (one who will do to start a progress, swell a train or two) — I found it a mite patronizing. The day when someone gives me an academic job solely because I occasionally make interesting comments on people’s work, streamline proofs, and every now and again solve a problem for someone, is the day when I will take this March-of-Progress rhetoric a little more seriously.

Publish-or-perish is not a mindset for many of us. It is an initial condition of the system we’ve found ourselves in. Also a Red Queen effect in place: I wouldn’t publish middling stuff, were it not for the fact that everyone else is (and they feel the same about me, I’m sure).

Oh, and actually writing things up well is either a lot harder than most people think, or something people have insufficient initiative to do well.

After re-reading my last post, I can see how my comment(s) may have offended you. My apologies are offered to all of you. But still, let’s desensitize our emotions for a second because progress is at stake. Certainly mine aren’t the only ideas being floated at the moment. However, bickering aside, they tend to be the ones that offer an immediate solution. If you have better ideas, or any, please share them.

Link Starbureiy, on his/her website (hover over the name to see it) describes itself as “Emperor of Dreams”.

Dear “Emperor of Dreams”,

I am not your subject.

Sincerely,
Nathaniel.

====
I think that “Emperor of Dreams” may have some sort of idea that “Progress” is a good thing, and should be pursued regardless of price. I disagree, and would counter by asserting that there are always risks to progress, and that it’s better to take a calm and balanced attitude on the subject.

Don’t be ridiculous! I find it disturbing to read your follow-up comment immediately after Michael Nielsen applauded the contributors to this blog (in particular) for having sense and sensibility.

Regardless, this Universe operates on very basic and simple principles of economics. They are:

1) There is risk in endeavor.
2) Cost is associated with any risk.
3) You pay for everything you do.

For one to imply that progress is “too risky” is preposterous. Dr. W.T. Gowers certainly isn’t the first person to suggest open collaboration on grand scales, by any means. Likewise, at this rate, and with this trite lack of enthusiasm, he won’t be the last.

For you to call me out on my moniker is low-class. Nevertheless, since it is something I pride myself on, you’re welcome to read my little blurb about it; hosted, ironically, on Google Knol (c & p –> http://knol.google.com/k/link-starbureiy/emperor-of-dreams/keic9mdr861/71# to your browser’s URL). One of my criticisms of GK is its current entry URL shortcumming (compare with the succinctness of Wikipedia).

As for your earlier question to whether I have a complete solution addressing Gowers’ task of assigning credit … well, yes and no. I’ll stubbornly insist that co-authors use their real identities instead of pseudonyms (e.g., Polymath1) during the collaboration. This is a prerequisite on Google Knol, anyway. Again, I foresee some new software being created to handle credit distribution. Someone must be making substantial and functional contributions in the collaboration cloud, so to speak. Because that issue has yet to be dealt with, I will also answer this question in the negative.

Before we go any further, I’d like to call attention to my snide attempt at perverted humor (“shortcummings”). I hope I don’t end up regretting it, as it won’t happen again.

In due time, I will move this conversation over to Google Knol. If you’re paying attention, most of my knols are ‘blank’. Basically, I’m using the service as a poor man’s copyright for right now. Google has recently been discontinuing programs of theirs (such as Notebook, Dodgeball, and others) because of lack of popularity, and rumormill has it that closing Knol could be in the future. Which brings me to this blog (gowers.wordpress.com). I figure that if they get enough heavy hitters and professionals using their service, then it stands a better chance of survival. Besides, I really believe it has huge potential to be the platform we’re trying to build.

gowers Says: *how one is supposed to evaluate a chemist, for example, who works as a member of a large team in a laboratory, for hiring purposes.*

Some journals detail the contributions of each author. Reference letters can help, e.g. with senior co-authors explaining the role played by junior co-authors. Failing such information, hiring committees count papers. Within a given field, the scaling factor (number of authors per paper) should be relatively constant. As for comparing across fields, the central challenge for tenure/award committees, there is often a reliance upon recommendation letters from departments, which explain issues of authorship.

Failing such things, committees simply count beans (papers). How should the count be adjusted for number $N_i$ on each $i$th paper? The simple answer is to sum $1/N_i$ across $i$, but that answer causes a lot of discussion in such committee meetings, as people who publish with a lot of co-authors dislike the idea that their paper count is inflated.

I do wish this system were threaded … beyond about 6 posts, a linear list becomes overwhelming.

If anyone wants to talk mathematical problem solving in real time chat you can try entering a discussion on the IRC channel on freenode #math

Here is a printout of searchIRC.com….

irc:// #math

Users: Current: 386, Avg: 361, Max: 446
Network: freenode

Directory:
Topic:
Education and Schools > Education Discussions
All aspects of mathematics (except software: join #math-software) | Channel URL: http://www.freenode-math.org | Don’t ask if you can ask a question, if anyone can help, or if anyone knows about . In addition, we are _NOT_ calculators. | Off topic? Take it to #not-math (we’re serious, the channel name is not a joke) | LaTeX paste: http://www.mathbin.net | Next seminar: TBD

@ Andrea’s comment: “This comment could range from “I learnt about this” or “I studied that paper” to “here’s an idea that could work for what I’m doing, I’m testing it next days” or “I finally have a proof of whatever, let me write it here”.”

Well, I suppose that microblogging could work just fine in this case. Maybe creating a group at twitter and see what happens. People could just post what you’ve mentioned: “Hi, today I was wondering about this problem and…” and so on. 🙂

[…] Is massively collaborative mathematics possible? [via nielsen] One day after Michael Nielsen’s post looking at how blogging creates a new forum for solving scientific problems, Fields Medal-winner Timothy Gowers decides “to suggest a problem and see what happens.” […]

[…] Timothy Gowers, a prominent mathematician, was inspired by Michael Nielsen’s post, to muse in this blog post about whether massively collaborated mathematics is possible. The post was later critiqued by […]

It is notable that most of the USA Mathematical Academia “poster-boys names” (except for Terence Tao ) choose not to take part in this discussion. They don’t read other’s blog, they don’t read emails from others – they busy … but with hat ?

[…] months ago Tim Gowers put forward the challenge of whether massively collaborative mathematics was possible. He also came up with a suitable problem and started it as a wiki. As well as the wiki and […]

[…] background to all this is that I’ve been collecting some thoughts about the ongoing Polymath project, an experiment in open source mathematics, and the question of how projects like Polymath can be […]

[…] Gowers has been running a remarkable experiment in how mathematics is done, a project he dubbed the Polymath1 project. Using principles similar to those employed in open source programming projects, he used blogs and […]

[…] to solve a long-standing mathematical problem in a massively collaborative way. It is called the Polymath1 Project and looks to provide a new proof of the density Hales-Jewett (DHJ) theorem. Apparently, after about […]

[…] to solve a long-standing mathematical problem in a massively collaborative way. It is called the Polymath1 Project and looks to provide a new proof of the density Hales-Jewett (DHJ) theorem. Apparently, after about […]

[…] experiement here. Recently some high profile mathematicians have been using a blog to conduct a massively collaborative mathematics project. What a brilliant idea that could be done today, with our current […]

[…] looks as though it were quite successful. This of course, answers Tim’s original question “is massively collaborative mathematics possible?” positively, but I still have to wonder if it’s sustainable in the long term. Of course, it […]

[…] blog, but he brought an interesting new spin to it in his presentation — he focused on the Polymath Project, a recent case in the math blogging community where the blogger Tim Gowers used his blog as a way […]

[…] as wikis, online forums and their descendants.” In a similar vein, Timothy Gowers started the Polymath project with a blog post discussing the following idea: It seems to me that, at least in theory, a […]

[…] Anyway, this example is perhaps a bit strained (and maybe it owes more to thermodynamics than to biology), but already it suggests a new mathematical object of study, namely the partition function as above, and one is already inclined to look for examples for which the partition function obeys a symmetry like that enjoyed by the Riemann zeta function, or to specialize temperature to other values, as in random matrix theory. The introduction of new methods into the study of a classical object — for example, the decision to use thermodynamic methods to organize the study of van Kampen diagrams — bends the focus of the investigation towards those examples and contexts where the methods and tools are most informative. Phenomena familiar in one context (power laws, frequency locking, phase transitions etc.) suggest new questions and modes of enquiry in another. Uninspired or predictable research programs can benefit tremendously from such infusions, whether the new methods are borrowed from other intellectual disciplines (biology, physics), or depend on new technology (computers), or new methods of indexing (google) or collaboration (polymath). […]

[…] methods), and the problem is receptive to the incremental, one-trivial-observation-at-a-time polymath approach. So I would like to invite people to try solving the problem collaboratively on this blog, by […]

[…] project announced – deterministic way to find primes New polymath project – massively collaborative mathematics project announced at the polymath blog – deterministic way to find primes – given an integer k, […]

[…] You may remember the polymath1 project, in which Tim Gowers envisioned, and realized, a “massively collaborative” approach to solving an open question in mathematics. The project succeeded, and in fact […]

[…] definitely check it out here. This all started with Terence Tao and Tim Gower’s ideas to try massively collaborative mathematics and the results were four incredibly active discussions. As I understand it, the idea is to develop […]

[…] in mathematics By Peter Cameron In October 2009, Nature published an account of a remarkable experiment by Tim Gowers. On his blog, he proposed that his readers should collaborate on a research project, […]

[…] Tim Gowers, an equally accomplished mathematician, has a blog on which he too discuses , ideas, thoughts, conjectures, and – like Terry Tao – exposes his ways of thinking and working. Tim has pushed the collaborative nature of mathematics further than anyone with his massively collaborative mathematics project. […]

[…] run to see how useful average people could be in complicated projects. One good example involved a crowd-sourced effort to solve a complex mathematical problem. The problem involved a board game which had all the same rules, only with many extra dimensions […]

[…] Gowers’ blog regularly had thousands of readers, including many of the world’s top mathematicians, so the blog thread soon had thousands of words and dozens of top mathematical thinkers participating. Six weeks later, the theorem was proven and the proof will be submitted to a top math journal, under the collective name “D.H.J. Polymath”. The Nature article describes a creative process just like the one that creativity researchers have identified, of creativity as a series of small insights, as described in my book Group Genius: For the first time one can see on full display a complete account of how a serious mathematical result was discovered. It shows vividly how ideas grow, change, improve and are discarded, and how advances in understanding may come not in a single giant leap, but through the aggregation and refinement of many smaller insights. […]

The polymath idea is excellent. I hope this catches on. I would like to see similar efforts by chemists/biologists etc in which experimentalist work with theoreticians to solve problems. I also would like to see continuous stream of comments after a paper is published. Also, reviewers comments should be posted without revealing their names.

[…] and invited the entire world to collaborate with him in proving his opinion correct. Following an initial post asking “Is Massively Collaborative Mathematics Possible?”, he posted a description of […]

[…] Rehmeyer of Science News has what I think is one of the best versions of the story of the Polymath Project — mathematician Tim Gowers’s successful experiment to solve a non-trivial mathematical […]

[…] January of 2009, Tim Gowers initiated an experiment in massively collaborative mathematics, the Polymath Project. The initial stage of this project was extremely successful, and led to two scientific papers: […]

[…] by sharing one’s insights, even if they are small or end up being inconclusive. (See also Tim Gowers’ original post regarding polymath projects.) But perhaps some tweaking to the rules may be needed. (For […]

[…] ongoing Polymath projects. Perhaps Polymath has not yet achieved Gowers’s original aim of massively collaborative mathematics, but it I don’t think that I’m the only person out there who thought that somewhat […]

HR issues aside, I think this sounds like a promising idea. It sounds as if a paradigm-shift of the notion of “worth” of a mathematician wight be in order for it to work; if it would become the case that this practice is widespread then the HR moguls will have to deal with that themselves.

[…] The other major innovation, Polymath, was the brainchild of a single person with a blog. Tim Gowers imagined the possibility, set out the rules and started the first project; other participating blogs and wikis were added in […]

[…] by sharing one’s insights, even if they are small or end up being inconclusive. (See also Tim Gowers’ original post regarding polymath projects.) But perhaps some tweaking to the rules may be needed. (For […]

1. This reminds open source, but partially. In particular, open source is a large set of problems, where people can find what they like and where they can contribute. Therefore, by probability argument the quality and volume constantly grows. It is of cause, is driven by few people, or teams, with sometimes money devoted for the project.

2. Wikipedia is a good reference, but this is passive, as the lectures posted on the web. What is missing is the structured system, where one can acquire skills. For example, a set of lectures associated with a set of exercised, where one can solve problem online, with the computer/server/program can evaluate solution (which can be formalized), and figure out the lack of knowledge in particular topic required to understand underlying concept. The next step for the course will be to present/review/test the missing topic. The advantages are – once written it can be used anyone who wants to learn/master certain topic. the task is divided among many professionals, and active people can gain skills with their own pace.

3. This two approaches combined lead, in a sense, to automated open source/wikipedia like knowledge base with skills learning. By the combined approach I mean that new knowledge is now required, in addition to presenting/expressing findings, encoding it in the form that people can acquire as a skill, with wrong understanding being corrected on the way (which is an efficient way to replace publications, ensure public review and refining). That at least speed up the acquisition for people new in the field, and ensure the quality of the knowledge, possible creating a certificate of understanding for particular person.

The revision system can be used to credit contributions. People have to be payed by the real contribution to the useful approaches, and particular work done – that will reinforce sharing ideas, at the end you are payed for what is taken. The wrong approaches are useful here in exercise creation/evaluation, and is also valuable. Think here of efficiencies. The downside is that huge cultural shift, especially with funding agencies are required.

This is a fantastic idea. I see the chief advantage being a sort of large assortment of ideas in which none become oppresive because they are presented in the spirit of whatever they’re worth. Mathematics suffers in my view because the publications so often exclude the underlying intuitions. So even if the project fails to advance a contribution, we need more public displays of the sort of struggling the researcher does in order to make something new.

[…] and others want to overthrow that model. Their shining example is this blog post by Tim Gowers. Gowers is a mathematician who proposed attacking an open math problem right there on his blog, by […]

[…] inviting other people to weigh in with their own suggestions for resolving them. He dubbed it the Polymath Project, an undertaking that ultimately produced a series of new ideas and insights as well as several […]

As part of my ongoing research to explore models for online collaboration I have recently been reviewing the Polymath project1 which exploits collaborative principals in resolving mathematical problems. The Polymath project created by the British …

[…] say, can do everything from mapping crime to funding wacky art projects to solving the world’s toughest mathematical conundrums. But even those who tout the wisdom of crowds for tasks that can be broken up into pieces, or for […]

[…] vague like “Science and the Internet,” in which I talked about twitter, RSS feeds, Tim Gower’s Polymath Project and some other things. Needless to say, I felt pretty gratified when this guy used the same example […]

[…] points to the huge (and somwhat surprising) success of Cambridge mathematician Timothy Gower’s Polymath project which invites mathematicians, be they professional or amateur, to contribute to cracking […]

[…] have made real progress in scientific research, including one called The Polymath Project, which started with a simple blog post by a mathematician at Cambridge University who wanted to see if he could get help with a problem. Within a matter of hours, comments had […]

[…] have made real progress in scientific research, including one called The Polymath Project, which started with a simple blog post by a mathematician at Cambridge University who wanted to see if he could get help with a problem. Within a matter of hours, comments had […]

[…] have made real progress in scientific research, including one called The Polymath Project, which started with a simple blog post by a mathematician at Cambridge University who wanted to see if he could get help with a problem. Within a matter of hours, comments had […]

[…] have made real progress in scientific research, including one called The Polymath Project, which started with a simple blog post by a mathematician at Cambridge University who wanted to see if he could get help with a problem. Within a matter of hours, comments had […]

[…] have made real progress in scientific research, including one called The Polymath Project, which started with a simple blog post by a mathematician at Cambridge University who wanted to see if he could get help with a problem. Within a matter of hours, comments had […]

[…] Professor Gowers is a British mathematician, Fields medalist 1998. Besides his many interesting results, Gowers has contributed to the mathematical community in many other valuable ways, for example, by prompting the creation and development of polymath projects, “massively collaborative mathematics“. […]

[…] (famous) mathematician, and Cambridge (UK) professor, who asked on his blog in 2009 whether science could be done collectively out in the open. There was a mathematical problem that he would like to solve, and so he set out via his blog to […]

[…] a 2009 post on his blog, Gowers asked the provocative question “is massively collaborative mathematics possible?” This post led to his creation of thePolymath Project, using the comment functionality of his blog […]

[…] the method the ‘Polymath Project’ (Nielsen 2009). The idea of ‘Polymath Project’ (Gowers 2009) implies that we can use Internet to build tools that actually expand our ability to solve the most […]

[…] just state that we should try to follow Timothy Gowers’s 12 ground rules at the end of his Polymath kick-off post. Also, I’ll credit Clark Alexander for walking me through this entire method, and […]

[…] the Polymath experience though. Back in 2009, Fields medallist Tim Gowers blogged about “massively collaborative mathematics“. He wrote: “The idea would be that anybody who had anything whatsoever to say about […]

[…] heard about it a few years ago. It was started by mathematician Tim Gowers in a blog post titled “Is massively collaborative mathematics possible?” [2] He and Terry Tao are the leaders of the project. They’re among the world’s top […]

[…] idea, that massively collaborative mathematics could be possible. Tim Gowers decide to announce an intellectual invitation on his WordPress blog, encouraging mathematicians the world over to help him openly solve a complex, unsolved problem by […]

[…] the authors of a number of papers which have arisen as a result of the polymath project initated by Gowers. Presumably, since it is a matter of open record, one can go through and identify the participants […]

[…] of course, if I go down the public route, it gives me another chance to try to promote the Polymathematical way of doing research, which on general grounds I think ought to be far more efficient. This is a strong additional […]

[…] has been doing polymath projects for years now. It all kicked off when Gowers posted the question, is massively collaborative mathematics possible? It’s a great post full of great ideas, and it turns out the answer is a […]

[…] to maybe get an idea/understanding, to apply OST to the some ideas expressed on this forum, grok? Is massively collaborative mathematics possible? It seems to me that, at least in theory, a different model could work: different, that is, from […]

[…] can do everything from mapping crime to funding wacky art projects to solving the world’s toughest mathematical conundrums. But even those who tout the wisdom of crowds for tasks that can be broken up into pieces, or for […]

[…] more than two authors. In 2009 Tim Gowers (who also set my essay topic!) raised the possibility of massively collaborative mathematics, in which an online community of mathematicians would pool their efforts on a single problem. On […]

[…] Massively collaborative mathematics (via Terry Tao). I count myself an idealist when it comes to such ideas, just as the author of this post, and there are some well argumented points therein, but… my more recent economics background takes me back to earth. So here’s one main reason (there are others, linked particularly to the nature of problem chosen to be solved by way of such “massive collaboration”) why I think this will not work out (in Maths or any other science, for that matter, Econ included): the costs (particularly time and effort to follow such discussions, not to mention trusting the person– if at all– to monitor it all etc) would far outweigh the benefits. Unless the persons participating are far more efficient than the average (in time management & co) and, perhaps crucially, are not concerned with career building any longer… Somebody like Terry Tao perhaps, to keep it to Maths, though he does not seem overenthusiastic either :-). […]

[…] classic example of this approach is provided by Tim Gowers, who posted in his blog a mathematical question and within a matter of days, the commenters had solved it. This gave birth to the Polymath […]

[…] and other beside the point comments aside) peers reviewing each others contributions. The original Polymath Project is a powerful example of this. (Disclaimer: As a historian by origin I love the monograph, longue […]

[…] and other beside the point comments aside) peers reviewing each others contributions. The original Polymath Project is a powerful example of this. (Disclaimer: As a historian by origin I love the monograph, longue […]

Very interesting entry, Gowers. I have written two articles on what I call it Crowd-Authoring. You and your readers might be interested in these articles, which can be found here: https://crowdauthoring.wordpress.com/

[…] the potential role of blogging in a research project. Timothy Gowers has started a very interesting blog-based mathematics project in which the blog itself serves as the medium for collaborative discovery. I started wondering […]

[…] open a discussion for a general public. A classic example of this approach is Tim Gowers, who posted in his blog a mathematical question and in a matter of days the commenters had solved it. From this case, Polymath Project, an online […]

It is August 2017 and open research is still not a thing yet. I wrote a blogpost proposing a Github model and hopefully it can be of use to anyone reading this far into the comment section after scrolling through many pingback articles (and this adds to the list of the pingbacks too!):http://ypei.me/2017/08/mathematical_bazaar/

[…] about 10 years ago when Timothy Gowers, a mathematician at the University of Cambridge, wanted to find a way to facilitate massive online collaborations in mathematics. Work on Polymath problems is done publicly, and anyone can contribute. Recently, de Grey was […]

[…] about 10 years ago when Timothy Gowers, a mathematician at the University of Cambridge, wanted to find a way to facilitate massive online collaborations in mathematics. Work on Polymath problems is done publicly, and anyone can contribute. Recently, de Grey was […]

[…] about 10 years ago when Timothy Gowers, a mathematician at the University of Cambridge, wanted to find a way to facilitate massive online collaborations in mathematics. Work on Polymath problems is done publicly, and anyone can contribute. Recently, de Grey was […]

[…] 10 years in the past when Timothy Gowers, a mathematician on the College of Cambridge, wished to discover a technique to facilitate huge on-line collaborations in arithmetic. Work on Polymath issues is completed publicly, and anybody can contribute. Not too long ago, de […]

[…] about 10 years ago when Timothy Gowers, a mathematician at the University of Cambridge, wanted to find a way to facilitate massive online collaborations in mathematics. Work on Polymath problems is done publicly, and anyone can contribute. Recently, de Grey was […]

[…] about 10 years ago when Timothy Gowers, a mathematician at the University of Cambridge, wanted to find a way to facilitate massive online collaborations in mathematics. Work on Polymath problems is done publicly, and anyone can contribute. Recently, de Grey was […]

[…] about 10 years ago when Timothy Gowers, a mathematician at the University of Cambridge, wanted to find a way to facilitate massive online collaborations in mathematics. Work on Polymath problems is done publicly, and anyone can contribute. Recently, de Grey was […]

[…] about 10 years ago when Timothy Gowers, a mathematician at the University of Cambridge, wanted to find a way to facilitate massive online collaborations in mathematics. Work on Polymath problems is done publicly, and anyone can contribute. Recently, de Grey was […]

[…] about 10 years ago when Timothy Gowers, a mathematician at the University of Cambridge, wanted to find a way to facilitate massive online collaborations in mathematics. Work on Polymath problems is done publicly, and anyone can contribute. Recently, de Grey was […]

[…] a thought experiment by Tom Gowers titled, “Is massively collaborative mathematics possible?” on his blog, has ended up as one of the most successful online mathematics collaborations ever. It’s called […]

[…] are scores of examples for this crowdsourced science: a classic example is that of Tim Gowers, who posted in his blog a mathematical question and in a matter of days the commenters had solved it. This gave birth to the Polymath Project, an […]